Do we have two Q-learning update formulas?
The first one is the update rule that we use in the $Q$-learning algorithm.

The second one is the "definition" of $Q(s, a)$ values, although I would personally write it as follows, with an expectation around the reward, to also support cases where rewards might be non-deterministic;

$$Q(s, a) \doteq \mathbb{E} \left[ r(s, a) \right] + \gamma \max_a Q(s', a)$$

Here, the $Q(s', a)$ itself would also be similarly defined, and again that one is also assumed to be a "ground-truth" value.

In practice, when learning, we do not actually know what any of these $Q$-values exactly are; that's why we're doing learning in the first place, we're trying to learn what these values are! More precisely, when we wish to assign a new value to $Q(s, a)$, the definition tells us that we should use $Q(s', a)$ for that, but we can't because we don't know exactly what the correct $Q(s', a$) value is either. 

We do generally have some $Q(S_{t+1}, a)$ values, but they're generally going to be only approximations, resulting from the previous steps of our own learning process (or maybe even randomly initialised values if we only just started training!). There may also be an additional approximation error in stochastic environments, where the true $Q(s', a)$ value may be a weighted average over multiple different possible successor states $s'$, but during the training we only observed a single concrete successor state $S_{t+1}$.

So, we can follow the definition and use $Q(S_{t+1}, a)$ as an approximation of $Q(s', a)$, but we know that it's just an approximation and not entirely reliable. Therefore, instead of fully applying the definition and completely replacing our $Q(s, a)$ value based on it, we interpolate between our current estimate $Q(s, a)$ and the new one that we should have according to the definition + our approximations. We only slightly shift towards it. The learning rate $\alpha$ determines how much we shift towards it. Normally we use a learning rate $0 &lt; \alpha &lt; 1$. Note that if you were to use $\alpha = 1$, you would actually recover the definition for $Q$-values again (except it would still use some observations to estimate unknown quantities).